Mixed effects models

An example

Single regression line

Known sources of variation

Variation in intercepts

Variation in intercepts and slopes

Variation in parameters

• in some cases, variation is directly of interest
• in other cases, it’s a nuisance
  • can break independence assumptions
  • can introduce extra noise
• mixed models can help

Mixed models

• mix of fixed and random effects
• these terms are not consistently defined
• in this context, really only matters for factors (categorical variables)

Fixed effects

• these are what we’ve used in general linear models
  • intercepts
  • slopes
  • interactions
• my definition: categories are independent

Random effects

• parameters differ among categories but categories aren’t fully independent
• some definitions:
  • random if we haven’t sampled the entire population
  • random if we “don’t care” about the factor
  • random if there is some form of shrinkage

Random effects

• examples: repeated measures, spatial blocks
  • can be a really good way to account for non-independent observations
• caveat: lme4 methods often require >5 levels for random effects models to work
• be pragmatic (and check model fit!)

Mixed models

• assumptions: much the same as a general linear model
• residuals are independent
• normally distributed residuals
• constant variance of residuals

Mixed models in R

• use a formula interface to define models

Mixed models in R

# load lme4 package
library(lme4)

# fit model with single intercept and slope
mod_lm <- lm(response ~ predictor)

# fit model with random intercepts
mod_int <- lmer(response ~ predictor + (1 | block))

# fit model with random intercepts and slopes
mod_slope <- lmer(response ~ predictor + (1 + predictor | block))

# fit model with nested random intercepts
mod_slope <- lmer(response ~ predictor + (1 + predictor | block / nested_block))

plot function

summary function

## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ x + (1 | z)
## 
## REML criterion at convergence: 2593.1
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.9891 -0.6603 -0.0267  0.6876  3.2835 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  z        (Intercept) 12.82    3.580   
##  Residual             10.03    3.166   
## Number of obs: 500, groups:  z, 5
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept) 100.0136     1.6074  62.222
## x            -0.3119     0.1450  -2.151
## 
## Correlation of Fixed Effects:
##   (Intr)
## x 0.004

Mixed models in R

# print the fixed effects
fixef(mod_int)

## (Intercept)           x 
## 100.0135534  -0.3118507

# print the random effects
ranef(mod_int)

## $z
##   (Intercept)
## 1    1.387692
## 2    5.573009
## 3   -2.021700
## 4   -1.574244
## 5   -3.364757

Mixed models in R

# print the fixed effects
fixef(mod_slope)

## (Intercept)           x 
##  99.9562825  -0.2379978

# print the random effects
ranef(mod_slope)

## $z
##   (Intercept)          x
## 1    1.317356 -1.4299245
## 2    5.629131 -0.7534335
## 3   -2.060671  0.7314980
## 4   -1.563628 -0.3570966
## 5   -3.322187  1.8089566

Interpreting random effects

• in short: don’t!
• if you care about it, it might be better as a fixed effect
• however, can still look at “variance components”
  • technical term: variance partitioning
• VarCorr(mod) is useful for this (but so is summary(mod))

Model assessment and model selection

• many different approaches (see Worksheet 1)
• start by assessing model fit
• but also need to assess model fit for purpose
• which model is “best”?
• my approach: often decide on random effects a priori and don’t “select” these